) The following is a proposed proof of the given statement. 1. Suppose m is any even integer and n is any odd integer. 2. If m·n is even, then by definition of even there exists an integer r such that m.n= 2r. 3. Also since m is even, there exists an integer p such that m = 2p by definition of even. 4. And since n is odd, there exists an integer q such that n = 2g + 1 by definition of odd. 5. Thus, by substitution, m·n = (2p)(2q + 1) = 2r, where ris an integer. 6. Hence, by definition of even, then, m.nis even, as was to be shown. Reorder the sentences in the following scrambled list to explain the mistake in the proposed proof. Step 2 states that the truth of S would follow from the assumption that m.n is even. Hence, the proposed proof is circular; it assumes what is to be proved. Step 5 deduces a conclusion that would be true if S were known to be true. Let S be the sentence, There is an integer r such that m n equals 2r." Thus, the conclusion in Step 5 is not a valid deduction. The truth of S is not known at the point where Step 5 occurs because the assumption that m n is even has not been proved. Since the truth of Step 6 depends on the truth of the conclusion in Step 5, Step 6 is not a valid deduction.

) The following is a proposed proof of the given statement. 1. Suppose m is any even integer and n is any odd integer. 2. If m·n is even, then by definition of even there exists an integer r such that m.n= 2r. 3. Also since m is even, there exists an integer p such that m = 2p by definition of even. 4. And since n is odd, there exists an integer q such that n = 2g + 1 by definition of odd. 5. Thus, by substitution, m·n = (2p)(2q + 1) = 2r, where ris an integer. 6. Hence, by definition of even, then, m.nis even, as was to be shown. Reorder the sentences in the following scrambled list to explain the mistake in the proposed proof. Step 2 states that the truth of S would follow from the assumption that m.n is even. Hence, the proposed proof is circular; it assumes what is to be proved. Step 5 deduces a conclusion that would be true if S were known to be true. Let S be the sentence, There is an integer r such that m n equals 2r." Thus, the conclusion in Step 5 is not a valid deduction. The truth of S is not known at the point where Step 5 occurs because the assumption that m n is even has not been proved. Since the truth of Step 6 depends on the truth of the conclusion in Step 5, Step 6 is not a valid deduction.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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